We prove the existence and uniqueness of the equilibrium state and conformal measure and the existence of a spectral gap for the transfer operator and its perturbations on various functional spaces

EXISTENCE AND PROPERTIES OF EQUILIBRIUM STATES OF HOLOMORPHIC ENDOMORPHISMS OF P^k

FABRIZIO BIANCHI AND TIEN-CUONG DINH

Abstract. We study the transfer (Perron-Frobenius) operator induced on P^k =P^k(C) by a generic holomorphic endomorphism and a suitable continuous weight. We prove the existence and uniqueness of the equilibrium state and conformal measure and the existence of a spectral gap for the transfer operator and its perturbations on various functional spaces. Moreover, we establish an equidistribution property for the backward orbits of points, with exponential speed of convergence, towards the conformal measure. Several statistical properties of the equilibrium state, such as the K-mixing, mixing of all orders, exponential mixing, ASIP, LIL, CLT, LDP, local CLT, almost sure CLT are also obtained. Our study in particular applies to the case of H¨older continuous weights and observables, and many results are already new in the case of zero weight function and even in dimension k = 1. Our approach is based on pluripotential theory and on the introduction of new invariant functional spaces in this mixed real-complex setting.

Contents

1. Introduction and results 2

2. Preliminaries and some comparison principles 6

3. Some semi-norms and equidistribution properties 13

4. Existence of the scaling ratio and equilibrium state 22

5. Spectral gap for the transfer operator 37

6. Statistical properties of equilibrium states 44

Appendix A. Abstract statistical theorems 53

References 57

Notation. Throughout the paper, P^k denotes the complex projective space of dimension k endowed with the standard Fubini-Study form ωFS. This is a K¨ahler (1,1)-form normalized so that ω^k_FS is a probability measure. We will use the metric and distance dist(·,·) on P^k induced by ωFS and the standard ones on C^k when we work on open subsets of C^k. We denote by B_P^k(a, r) (resp. B^k_r,D(a, r),Dr) the ball of center a and radius r in P^k (resp. the ball of center 0 and radiusr inC^k, the disc of centeraand radiusr inC, and the disc of center 0 and radius r inC). Leb denotes the standard Lebesgue measure on a Euclidean space or on a sphere. The currentsω_nand their dynamical potentials u_n are introduced in Section 3.1.

The pairing h·,·i is used for the integral of a function with respect to a measure or more generally the value of a current at a test form. IfS andR are two (1,1)-currents, we will write

|R| ≤S when <(ξR)≤S for every functionξ:P^k→Cwith|ξ| ≤1, i.e., all currentsS− <(ξR) with ξ as before are positive. Notice that this forces S to be real and positive. We also write other inequalities such as|R| ≤ |R₁|+|R₂|if|R| ≤S1+S2 whenever |R₁| ≤S1 and |R₂| ≤S2. Recall thatd^c= _2πⁱ (∂−∂) anddd^c= _πⁱ∂∂. The notations.and &stand for inequalities up to a multiplicative constant. The function identically equal to 1 is denoted by 1. We also use the function log^?(·) := 1 +|log(·)|.

Consider a holomorphic endomorphism f:P^k → P^k of algebraic degreed≥2 satisfying the Assumption (A) in the Introduction. Denote respectively by T, µ = T^k, supp(µ) the Green (1,1)-current, the measure of maximal entropy (also called the Green measure or the equilibrium

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measure), and the small Julia set off. IfSis a positive closed (1,1)-current onP^k, its dynamical potential is denoted byu_S and is defined in Section 2.2. Ifνis an invariant probability measure, we denote by Ent_f(ν) the metric entropy of ν with respect tof.

We also consider a weight φ which is a continuous function on P^k. We often assume that φ is real. The transfer operator (Perron-Frobenius operator) L = L_φ is introduced in the Introduction together with the scaling ratio λ = λ_φ, the conformal measure m_φ, the density function ρ = ρ_φ, the equilibrium state µ_φ = ρm_φ, the pressure P(φ), see also Section 4. The measuresm_φ and µ_φ are probability measures. The operator L is a suitable modification of L and is introduced in Section 4.5.

The oscillation Ω(·), the modulus of continuity m(·,·), the semi-norms k·k_logp and k·k_∗ of a function are defined in Section 2.1. Other norms and semi-norms k·k_p,k·k_p,α,k·k_hp,αi,k·k_hp,αi,γ for (1,1)-currents and functions are introduced in Section 3 and the normsk·k

1,k·k

2 in Section 5.3. The semi-norms we consider are almost norms: they vanish only on constant functions. It is easy to make them norms by adding a suitable functional such as g 7→ |hm_φ, gi|. However, for simplicity, it is more convenient to work directly with these semi-norms. The versions of these semi-norms for currents are actually norms. The positive numbersq0, q1, q2 are defined in Lemmas 3.10, 3.13, 3.16 and the families of weightsP(q, M,Ω),P₀(q, M,Ω),Q₀ are introduced in Sections 4.5 and 5.2.

1. Introduction and results

Letf:P^k→P^kbe a holomorphic endomorphism of the complex projective spaceP^k=P^k(C), with k ≥ 1, of algebraic degree d≥ 2. Denote by µ the unique measure of maximal entropy for the dynamical system (P^k, f) [Lyu83, BD09, DS10a]. The support supp(µ) of µ is called the small Julia set of f. The measure µ corresponds to the equilibrium state of the system in the case without weight, i.e., when the weight is zero. In this paper, we will consider the case where the weight, denoted byφ, is not necessarily equal to zero. This problem has been studied for H¨older continuous weights using a geometric approach, in dimension 1, see, e.g., Denker- Przytycki-Urba´nski [Prz90, DU91a, DU91b, DPU96] and Haydn [Hay99] just to name a few, and in higher dimensions, see Szostakiewicz-Urba´nski-Zdunik [UZ13, SUZ14]. We will develop here an analytic method which will allow us to obtain more general and more quantitative results. Many results are new even whenφ= 0 and even fork= 1.

Throughout this paper, we make use of the following technical assumption for f:

(A) the local degree of the iterate fⁿ:=f ◦ · · · ◦f (ntimes) satisfies

n→∞lim 1

nlog max

a∈P^kdeg(fⁿ, a) = 0.

Here, deg(fⁿ, a) is the multiplicity ofaas a solution of the equation fⁿ(z) =fⁿ(a). Note that generic endomorphisms of P^k satisfy this condition, see [DS10b]. Our study still holds under a weaker condition that the exceptional set of f (i.e., the maximal proper analytic subset of P^k invariant by f⁻¹) is empty or more generally has no intersection with supp(µ). However, this situation requires more technical conditions on the weightφ. We choose not to present this case here in order to simplify the notation and focus on the main new ideas introduced in this topic.

Our first goal in this paper is to prove the following theorem.

Theorem 1.1. Let f be an endomorphism of P^k of algebraic degree d≥ 2 and satisfying the Assumption (A)above. Let φ be a real-valued log^q-continuous function onP^k, for someq >2, such that Ω(φ) := maxφ−minφ <logd. Thenφ admits a unique equilibrium state µφ, whose support is equal to the small Julia set of f. This measure µ_φ is K-mixing and mixing of all orders. Moreover, there is a unique conformal measurem_φassociated toφ. We have µ_φ=ρm_φ for some strictly positive continuous functionρonP^kand the preimages of points byfⁿ(suitably weighted) are equidistributed with respect to mφ as n goes to infinity.

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We say that a function is log^q-continuous if its oscillation on a ball of radiusr is bounded by a constant times (log^?r)^−q, see Section 2.1 for details. See also Section 4.5 for the K-mixing and mixing of all orders.

An equilibrium state as in the statement above is defined as follows, see for instance [Rue72, Wal00, PU10]. Given aweight, i.e., a real-valued continuous function, φas above, we define the pressure of φas

P(φ) := sup

Entf(ν) +hν, φi ,

where the supremum is taken over all Borel f-invariant probability measures ν and Ent_f(ν) denotes the metric entropy of ν. An equilibrium state for φ is then an invariant probability measure µ_φrealizing a maximum in the above formula, that is,

P(φ) = Entf(µφ) +hµ_φ, φi.

On the other hand, aconformal measure is defined as follows. Define the Perron-Frobenius (or transfer) operator L with weightφas (we often drop the indexφfor simplicity)

(1.1) Lg(y) :=L_φg(y) := X

x∈f⁻¹(y)

e^φ(x)g(x),

where g: P^k → R is a continuous test function and the points x in the sum are counted with multiplicity. A conformal measure is an eigenvector for the dual operatorL^∗ acting on positive measures.

Notice that, in the case where φ is H¨older continuous, Theorem 1.1 was established by Urba´nski-Zdunik [UZ13], see also [Prz90, DU91a, DU91b, DPU96] for previous results in dimension k = 1. When φ is constant, the operator L reduces to a constant times the push- forward operator f∗ and we get µ_φ=µ. For an account of the known results in this case, see for instance [DS10a].

A reformulation of Theorem 1.1 is the following: given φ as in the statement, there exist a number λ > 0 and a continuous function ρ = ρ_φ:P^k → R such that, for every continuous functiong:P^k→R, the following uniform convergence holds:

(1.2) λ⁻ⁿLⁿg(y)→cgρ

for some constantcg depending on g. By duality, this is equivalent to the convergence, uniform on probability measures ν,

(1.3) λ⁻ⁿ(L^∗)ⁿν →mφ,

wherem_φis a conformal measure associated to the weight φ. The equilibrium stateµ_φ is then given byµ_φ=ρm_φ, and we havecg =hm_φ, gi.

To prove Theorem 1.1, in Section 4 we develop a new and completely different approach with respect to [UZ13] and to the previous studies in dimension 1. As we will see later, the flexibility of this method will allow for a more quantitative understanding of the convergences (1.2) and (1.3), and for the direct establishment of several statistical properties of the equilibrium states.

The main idea of our method is the following. Let us just consider for now the case where both of the functionsgand φare of classC² (the general case requires suitable approximations of g and φ by C² functions). Given such a function g, first we want to prove that the ratio between the maximum and the minimum ofLⁿgstays bounded withn. This allows us to define the good scaling ratioλand to get that the sequence λ⁻ⁿLⁿg is uniformly bounded. Next, we would like to prove that this sequence is actually equicontinuous. This, together with other technical arguments, would imply the existence and uniqueness of the limit function ρ.

In order to establish the above controls, we study the sequence of (1,1)-currents given by dd^cLⁿg. First we prove that suitably normalized versions of these currents are uniformly bounded by a common positive closed (1,1)-current R. This is the core of our method which replaces all controls on the distortion of inverse branches of fⁿ in the geometric method of

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[UZ13] by a unique, global, and flexible estimate. Namely, for every n ∈ N we can get an estimate of the form

(1.4)

dd^cLⁿg cn

.

∞

X

j=0

e^Ω(φ) d

j(f∗)^jωFS

d^(k−1)j with cn:=kgk_C2hω_FS^k ,Lⁿ1i.

Here, ωFS denotes the usual Fubini-Study form onP^k normalized so that ω_FS^k is a probability measure. Notice that the last infinite sum gives a key reason for the assumption Ω(φ) <logd made on the weightφ as the mass of the current (f∗)^jωFS is equal tod^(k−1)j.

We will establish in Section 2 some general criteria, interesting in themselves, which allow one to bound the oscillation ofc⁻¹_n Lⁿgin terms of the oscillation of the potentials of the current in the RHS of (1.4). This latter oscillation is actually controllable. Assumption (A) allows us to have a simple control which makes the estimates less technical but such a control exists without Assumption (A).

Combining all these ingredients, Theorem 1.1 then follows from standard arguments that we recall in Sections 4.5 and 4.6 for completeness. We also prove that the entropy of µφ is larger than klogd−Ω(φ) > (k−1) logd, and that all the Lyapunov exponents of µ_φ are strictly positive. This also leads to a lower bound for the Hausdorff dimension of µφ.

Without extra arguments, given a continuous test function g the convergence as n → ∞ in (1.2) is not uniform in g. Our next and main goal is to establish an exponential speed of convergence in (1.2). This requires to build a suitable (semi-)norm for (or equivalently, a suitable functional space on) which the operator λ⁻¹L turns out to be a contraction.

Establishing the following statement is then our main goal in the current paper. As far as we know, this is the first time that the existence of a spectral gap for the perturbed Perron-Frobenius operator is proved in this context even in dimension 1, except for hyperbolic endomorphisms or for weights with ad-hoc conditions (see for instance [Rue92, MS00]). This is one of the most desirable properties in dynamics.

Theorem 1.2. Let f, q, φ, ρ, λ, m_φ be as in Theorem 1.1 and L the above Perron-Frobenius operator associated to φ. Let A > 0 and 0 < Ω < logd be two constants. Then, for every constant 0 < γ ≤ 1, there exist two explicit equivalent norms for functions on P^k: k·k

1, depending on f, γ, q and independent of φ, andk·k

2, depending on f, φ, γ, q, such that k·k_∞+k·k_logq .k·k

1 ' k·k

2 .k·k_Cγ.

Moreover, there are positive constants c = c(f, γ, q, A,Ω) and β =β(f, γ, q, A,Ω) with β < 1, both independent of φand n, such that whenkφk

1 ≤A andΩ(φ)≤Ωwe have kλ⁻ⁿLⁿk₁ ≤c, kρk

1 ≤c, k1/ρk

1 ≤c, and

λ⁻¹Lg

2 ≤βkgk

2

for every function g:P^k → R with hm_φ, gi = 0. Furthermore, given any constant 0 < δ <

d^γ/(2γ+2), whenA is small enough, the norm k·k

2 can be chosen so that we can take β = 1/δ.

The construction of the norms k·k

1 and k·k

2 is quite involved. We use here ideas from the theory of interpolation between Banach spaces [Tri95] combined with techniques from pluripotential theory and complex dynamics. Roughly speaking, an idea from interpolation theory allows us to reduce the problem to the case where γ = 1. The definition of the above norms in this case requires a control of the derivatives of g (in the distributional sense), and this is where we use techniques from pluripotential theory. This also explains why these norms are bounded by the C¹ norm. Note that we should be able to bound the derivatives of Lg in a similar way. A quick expansion of the derivatives of Lg using (1.1) gives an idea of the difficulties that one faces.

Our solution to the problem is to define a new invariant functional space and norm in this mixed real-complex setting, that we call the dynamical Sobolev space and semi-norm, taking into account both the regularity of the function and the action of f, see Definitions 3.12 and 3.15. The construction of this norm requires the definition of several intermediate semi-norms

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and the precise study of the action of the operator f∗ with respect to them, and is carried out in Section 3. Some of the intermediate estimates already give new or more precise convergence properties for the operatorf∗ and the equilibrium measureµ, see for instance Theorem 3.6.

A spectral gap for the Perron-Frobenius operator and its perturbations is one of the most desirable properties in dynamics. It allows us to obtain several statistical properties of the equilibrium state. In the present setting, we have the following result, see Appendix A for the definitions.

Theorem 1.3. Let f, φ, µ_φ, m_φ,k·k

1 be as in Theorems 1.1 and 1.2 and assume that kφk

1 <

∞. Then the equilibrium state µφ is exponentially mixing for observables with bounded k·k

1

norm and the preimages of points by fⁿ (suitably weighted) equidistribute exponentially fast towardsm_φ asngoes to infinity. Moreover, the measureµ_φsatisfies the LDP for all observables with finitek·k

1 norm, the ASIP, CLT, almost sure CLT, LIL for all observables with finitek·k

1

norm which are not coboundaries, and the local CLT for all observables with finite k·k

1 norm which are not (k·k

1, φ)-multiplicative cocycles.

In particular, all the properties in Theorem 1.3 hold when the weight φ and the observable are H¨older continuous and satisfy the necessary coboundary/cocycles requirements. In this assumption, some of the above properties were previously obtained with ad hoc arguments, see [PUZ89, DU91a, DU91b, DPU96, Hay99, DNS07, SUZ15] when k= 1, [UZ13, SUZ14] for mixing, CLT, LIL whenk ≥1, and [Dup10] for the ASIP when k≥1 and φ= 0. Our results are more general, sharper, and with better error control. Note that the LDP and the local CLT are new even for φ= 0 (for all k≥1 and for all k >1, respectively). Our method of proof of Theorem 1.3 exploits the spectral gap established in Theorem 1.2 and is based on the theory of perturbed operators. This approach was first developed by Nagaev [Nag57] in the context of Markov chains, see also [RE83, Bro96, Gou15], and provides a unified treatment for all the statistical study.

For the reader’s convenience, the above statistic properties (exponential mixing, LDP, ASIP, CLT, almost sure CLT, LIL, local CLT) and the notions of coboundary and multiplicative cocycle will be recalled in Section 6 and Appendix A at the end of the paper.

Outline of the organization of the paper. In Section 2, we introduce some useful notions and establish comparison principles for currents and potentials that will be the technical key to prove Theorems 1.1 and 1.2. In Section 3 we introduce the main (semi-)norms that we will need, and study the action of the operator f∗ with respect to these (semi-)norms. Section 4 is dedicated to the proof of Theorem 4.1. For this purpose, we develop our method to get the uniform boundedness and equicontinuity for the sequence Lⁿg, properly normalized, that lead to the good definition of the scaling ratioλ. Once this is done, the deduction of Theorem 1.1 is classical, and we just recall the proof of this in Sections 4.5 and 4.6 for the reader’s convenience. Section 5 is dedicated to the proof of Theorem 1.2, which is built on the method developed in Section 4, made quantitative with respect to the semi-norms that we introduced in Section 3. Finally, in Section 6 we develop the statistical study of the equilibrium states.

This section contains the proof of Theorem 1.3 and more precise statements. In Appendix A we recall statistical properties and criteria in abstract settings that we use to prove results in Section 6.

Acknowledgements. The first author would like to thank the National University of Singapore (NUS) for its support and hospitality during the visits where this project started and developed, and Imperial College London were he was based during the first part of this work. The paper was partially written during the visit of the authors to the University of Cologne. They would like to thank this university, Alexander von Humboldt Foundation, and George Marinescu for their support and hospitality.

This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sk lodowska-Curie grant agreement No 796004, the

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French government through the Programme Investissement d’Avenir (I-SITE ULNE / ANR- 16-IDEX-0004 ULNE and LabEx CEMPI /ANR-11-LABX-0007-01) managed by the Agence Nationale de la Recherche, the CNRS through the program PEPS JCJC 2019, and the NUS through the grants C-146-000-047-001 and R-146-000-248-114.

2. Preliminaries and some comparison principles

2.1. Some definitions. We collect here some notions that we will use throughout the paper.

Definition 2.1. Given a subsetU ofP^k orC^k and a real-valued functiong:U →R, define the oscillation ΩU(g) of gas

ΩU(g) := supg−infg and its continuity modulus m_U(g, r) at distancer as

m_U(g, r) := sup

x,y∈U: dist(x,y)≤r

|g(x)−g(y)|.

We may drop the indexU when there is no possible confusion.

The following semi-norms will be the main building blocks for all the semi-norms that we will construct and study in the sequel.

Definition 2.2. The semi-normk·k_logp is defined for every p >0 andg:P^k→R as kgk_logp := sup

a,b∈P^k

|g(a)−g(b)| ·(log^?dist(a, b))^p = sup

r>0,a∈P^k

Ω_B

Pk(a,r)(g)·(1 +|logr|)^p, whereB_Pk(a, r) denotes the ball of centeraand radiusr inP^k.

Definition 2.3. The normk·k_∗ of a (1,1)-currentR is given by kRk_∗ := infkSk

where the infimum is taken over all positive closed (1,1)-currents S such that |R| ≤ S, see the Notation at the beginning of the paper. When such a current S does not exist, we put kRk_∗ := +∞. The semi-norm k·k_∗ of an integrable functiong:P^k→R is given by

kgk_∗ :=kdd^cgk_∗.

Note that when R is a real closed (1,1)-current the above norm is equivalent to the usual one defined as

kRk_∗ := inf(kS⁺k+kS⁻k)

where the infimum is taken over all positive closed (1,1)-currents S^± on P^k such that R = S⁺−S⁻. In particular, forR=dd^cg, the currentsS⁺and S⁻are cohomologous and thus have the same mass, i.e., kS⁺k=kS⁻k, see [DS10a, App. A.4] for details.

In this paper we only consider continuous functions g. So the above semi-norms (and the others that we will introduce later) are almost norms: they only vanish when g is constant.

In particular, they are norms on the space of functions g satisfying hν, gi = 0 for some fixed probability measure ν. We will use later ν = mφ or ν = µφ to obtain a spectral gap for the Perron-Frobenius operator and to study the statistical properties ofµ_φ.

2.2. Dynamical potentials. LetT denote the Green (1,1)-current off. It is positive closed and of unit mass. LetSbe any positive closed (1,1)-current of massmonP^k. There is a unique functionu_S:P^k→R∪ {−∞} which is p.s.h. modulomT and such that

S=mT +dd^cu_S and hµ, u_Si= 0.

Locally, u_S is the difference between a potential of S and a potential of mT. We call it the dynamical potential of S. Observe that the dynamical potential of T is zero, i.e., uT = 0.

Recall that T has H¨older continuous potentials. So, u_S is locally the difference between a p.s.h. function and a H¨older continuous one. The dynamical potential ofS behaves well under

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the push-forward and pull-back operators associated to f. Indeed, because of the invariance properties of T, we have

f^∗S =md·T+dd^c(uS◦f) and f∗S =md^k−1·T+dd^c(f∗uS), which, together with the invariance properties ofµ, imply

uf^∗S =uS◦f and uf∗S=f∗uS.

We refer the reader to [DS10a] for details. In this paper, we only need currentsS such thatu_S is continuous.

2.3. Comparisons between currents and their potentials. A technical key point in the proof of our main theorems will be based on the following general idea: if u and v are two functions on some domain in C^k such that |dd^cu| ≤ dd^cv ori∂u∧∂u ≤ dd^cv, then u inherits some of the regularity properties ofv. This section and the next are devoted to make this idea precise and quantitative for our purposes. We start with the simplest occurrence of this fact in the first case in terms of the sup-norm.

Lemma 2.4. There exists a positive constant A such that, for every positive closed (1,1)- current S₀ on P^k of mass 1 and for every positive closed (1,1)-current S on P^k with S ≤ S₀, we have Ω(uS) ≤A+ Ω(uS0), whereuS0 and uS denote the dynamical potentials of S0 and S, respectively.

Proof. We assume that Ω(u_S0) is finite, since otherwise the assertion trivially holds. Observe that the mass m ofS is at most equal to 1 because S≤S0. Recall thatuS and uS0 satisfy

S =mT +dd^cu_S, S₀=T+dd^cu_S0, hµ, u_Si= 0, and hµ, u_S0i= 0.

The last identity implies that supu_S0 is non-negative.

We first prove that uS is bounded above by a constant. As mentioned above, the correspondence between positive closed (1,1)-currents and their dynamical potentials is a bijection. Moreover, we know that quasi-p.s.h. functions (i.e., functions that are locally difference between a p.s.h. and a smooth function) are integrable with respect toµ[DS10a, Th. 1.35]. Since the set of positive closed (1,1)-currents of mass less than or equal to 1 is compact, uS belongs to a compact family of p.s.h. functions modulo mT. We deduce that there is a constant A >0 independent ofS such thatu_S ≤A/2 on P^k, see [DS10a, App. A.2] for more details. It follows that supuS ≤supuS0+A/2 because supuS0 is non-negative.

Consider the current S⁰ := S₀ −S which is positive closed and smaller than S₀. By the uniqueness of the dynamical potential, we haveu_S⁰ =uS0 −uS, which implies uS =uS0−u_S⁰. Since S⁰≤S, as above, we also have supu_S⁰ ≤A/2. It follows that

infuS ≥infuS0−supuS⁰ ≥infuS0 −A/2.

This estimate and the above inequality supu_S ≤supu_S0 +A/2 imply the lemma.

Corollary 2.5. There exists a positive constant A such that for every positive closed (1,1)- current S₀ on P^k and for every continuous function g: P^k → R with |dd^cg| ≤ S₀ we have Ω(g)≤AkS₀k+ 3Ω(u_S0).

Proof. By linearity we can assume that S0 is of mass 1/2. Define R:= dd^cg and write it as a difference of positive closed currents, R = (R+S₀)−S₀. Since R+S₀ and S₀ belong to the same cohomology class, they have the same mass 1/2. We denote as usual by uR+S0 and uS0

the dynamical potentials ofR+S₀ and S₀ respectively.

A direct computation gives dd^c(g−u_R+S0+u_S0) = 0 which implies thatg−u_R+S0 +u_S0 is a constant function. Thus,

Ω(g) = Ω(u_R+S0 −u_S0)≤Ω(u_R+S0) + Ω(u_S0).

The assertion follows from Lemma 2.4 applied toR+S0,2S0 instead of S, S0. We use here the fact thatR+S₀ =dd^cg+S₀≤2S₀ and that 2S₀ is of mass 1. We also use a constantA which

is equal to twice the one in Lemma 2.4.

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The following result gives a quantitative control on the oscillation of u in terms of the oscillation of v. Notice in particular that it implies that, if v is H¨older or log^p-continuous for some p > 0, thenu enjoys the same property with possibly a loss in the H¨older exponent, but not in the log^p-exponent.

Proposition 2.6. Let u and v be two p.s.h. functions on B^k₃ such that dd^cu ≤ dd^cv and v is continuous. Then u is continuous and for every 0 < s ≤ 1 there is a positive constant A (independent of u and v) such that, for every 0< r≤1/2, we have

m_B^k

1(u, r)≤m_B^k

2(v, r^s) +Am_B^k

2(u, r^s)r^1−s ≤m_B^k

2(v, r^s) +AΩ_B^k

2(u)r^1−s.

Proof. The continuity of u is a well-known property. Indeed, since dd^cv−dd^cu is a positive closed (1,1)-current, there is a p.s.h. functionu⁰ such thatdd^cu⁰=dd^cv−dd^cu. So, bothu+u⁰ and v are potentials of dd^cv. We deduce that they differ by a pluriharmonic function. Hence u+u⁰ is continuous. We then easily deduce that bothuand u⁰ are continuous because both are p.s.h. (and hence u.s.c.).

We prove now the estimate in the lemma. Letx, y∈B^k₁ be such thatkx−yk ≤r. We need to boundu(y)−u(x). Without loss of generality, we can reduce the problem to the casek= 1 by restricting ourselves to the complex line throughx andy. Moreover, by translating and adding constants to u and v, we can assume thatx = 0, |y| ≤r,u(x) =v(x) = 0, and u(y)≥0. It is then enough to prove that

u(y)≤m_D1(v, r^s) +AΩ_Drs(u)r^1−s

for some positive constant Aand for u, v defined onD₂. Note that Ω_Drs(u)≤2m_D1(u, r^s).

Claim. We have, for some positive constantA, u(y)≤ 1

Leb(∂D_r^s) Z

|z|=r^s

u(z)dLeb(z) +AΩ_Drs(u)r^1−s.

Assuming the claim, we first complete the proof of the lemma. Let eu (resp. ev) be the radial subharmonic function on D2 such that u(z) (resp.e ev(z)) is equal to the mean value of u (resp.

v) on the circle of center 0 and radius|z|. Using the Claim, in order to obtain the lemma, it is enough to show thatue≤ev.

Recall that v −u is a subharmonic function vanishing at 0. Therefore, ev−eu is a radial subharmonic function vanishing at 0. Radial subharmonic functions are increasing in|z|. Thus, ev−euis a non-negative function and the lemma follows.

Proof of the Claim. Defineu⁰(z) :=u(zr^s) andy⁰ :=y/r^s. We need to show that, for|y⁰| ≤r^1−s, u⁰(y⁰)≤ 1

Leb(∂D₁) Z

∂D1

u⁰(z)dLeb(z) +AΩ_D1(u⁰)r^1−s.

We can assume, without loss of generality, that y⁰ = α ∈ R⁺ and α ≤ r^1−s. Consider the automorphism Ψ of the unit disc given by Ψ(z) = _1+αz^z+α. The map Ψ satisfies Ψ(0) = y⁰ and moreover Ψ extends smoothly to ∂D₁ and tends to the identity in the C¹ norm as α → 0. It follows that

Ψ^±1−id

_C1 ≤A⁰α≤A⁰r^1−s for some positive constant A⁰.

Define u⁰⁰:=u⁰◦Ψ and denote by ν the normalized standard Lebesgue measure on the unit circle. We deduce from the last inequalities that Ψ∗ν−ν is given by a smooth 1-form on ∂D₁ and kΨ_∗ν−νk_∞=O(r^1−s). Applying the submean inequality to the subharmonic functionu⁰⁰ we get

u⁰(y⁰) =u⁰⁰(0)≤ hν, u⁰⁰i=hν, u⁰◦Ψi=hΨ∗ν, u⁰i=hν, u⁰i+hΨ∗ν−ν, u⁰i.

Since Ψ∗ν and ν are probability measures, the integralhΨ_∗ν−ν, u⁰i does not change if we add tou⁰ a constant c. With the choice c=−infD1u⁰ (observe that u⁰ is continuous onD1) we get

u⁰(y⁰)≤ Z

∂D1

u⁰ dν+ sup

D1

|u⁰+c|O(r^1−s)≤ Z

∂D1

u⁰ dν+AΩD1(u⁰)r^1−s

for some positive constant A. This implies the desired inequality.

8

Corollary 2.7. Let v be a continuous p.s.h. function onB^k₃. Letu be a continuous real-valued function on B^k₃ such that|dd^cu| ≤dd^cv. Then for every0< s≤1 we have for 0< r≤1/2

m_Bk

1(u, r)≤3m_Bk

2(v, r^s) +A

Ω_Bk

2(u) + Ω_Bk 2(v)

r^1−s, where A is a positive constant independent of u andv.

Proof. Since |dd^cu| ≤ dd^cv, we have dd^c(u+v) = dd^cu+dd^cv ≥ 0. So the function u+v is p.s.h.; observe also thatdd^c(u+v) =dd^cu+dd^cv≤2dd^cv. Therefore, we can apply Proposition 2.6 tou+v,2v instead ofu, v. This gives

m_Bk

1(u, r)≤m_Bk

1(u+v, r) +m_Bk

1(v, r)≤m_Bk

2(2v, r^s) +AΩ_Bk

2(u+v)r^1−s+m_Bk 2(v, r)

≤3m_B^k

2(v, r^s) +A

Ω_B^k

2(u) + Ω_B^k

2(v)

r^1−s,

which is the desired estimate.

Corollary 2.8. Let S0 be a positive closed(1,1)-current onP^k with continuous local potentials.

Let F(S0) denote the set of all continuous real-valued functions g onP^k such that |dd^cg| ≤S0. Then F(S₀) is equicontinuous.

Proof. Letg be as in the statement. We coverP^k with a finite family of open sets of the form Φ_j(B^k_1/2) where Φ_j is an injective holomorphic map from B^k₄ toP^k. Write S₀ =dd^cv_j for some continuous p.s.h. function vj on Φj(B^k₄) and define Vj := Φj(B^k₃).

We apply Corollary 2.7 to g, v_j restricted toV_j instead of u,v and tos= 1/2. Taking into account the distortion of the maps Φ_j, we see that for all r smaller than some constant r₀ >0

m_Pk(g, r)≤3 max

j m_Vj(v_j, c√

r) +A

Ω_Pk(g) + max

j Ω_Vj(v_j)√ r,

where c ≥ 1 is a constant. Since Ω_Pk(g) is bounded by Corollary 2.5, the RHS of the last inequality is bounded by a constant _r which is independent of g and tends to 0 when r tends to 0. It is now clear that the familyF(S₀) is equicontinuous.

2.4. Further comparisons. In our study we will be naturally lead to also consider currents of the formi∂u∧∂u. These currents are always positive. In this section we study the regularity¯ of u under the assumption that i∂u∧∂u¯ ≤ dd^cv for some v of given regularity. Recall that, given a smooth bounded open set Ω⊂ C^k, the Sobolev space W^1,2(Ω) is defined as the space of functionsu: Ω→R such thatkuk_W1,2(Ω):=kuk_L2(Ω)+k∂uk_L2(Ω) <∞, where the reference measure is the standard Lebesgue measure on Ω. The Poincar´e-Wirtinger’s inequality implies thatkuk_W1,2(Ω).k∂uk_L2(Ω)+

R

Ωu dLeb

. We will need the following lemmas.

Lemma 2.9. There is a universal positive constant csuch that Z

K

|u|dLeb≤cLeb(K)(log^?Leb(K))^1/2.

for every compact set K⊂D₁ withLebK >0 and functionu:D₁→R such that kuk_W1,2(D1)≤ 1.

Proof. By Trudinger-Moser’s inequality [Mos71], there are positive constantsc₀andαsuch that Z

D1

e^2α|u|2dLeb≤c0.

Let m denote the restriction of the measure Leb to K multiplied by 1/Leb(K). This is a probability measure. It follows from Cauchy-Schwarz’s inequality that

Z

e^α|u|2dm≤Z

e^2α|u|2dm1/2

.Leb(K)^−1/2.

9

Observe that the function t 7→ e^αt2 is convex on R⁺ and its inverse is the function t 7→

α^−1/2(logt)^1/2. By Jensen’s inequality, we obtain Z

|u|dm≤α^−1/2 h

log Z

e^α|u|2dm i1/2

.(log^?Leb(K))^1/2.

The lemma follows.

Lemma 2.10. Let u:D2 → R be a continuous function and χ: D2 → R a smooth function with compact support in D₂ and equal to 1 on D₁. Set χ_z := ∂χ/∂z. Then we have, for all 0< r < s <1,

(2.1) u(0)−u(r) = 1 2π

D

i∂u, χ(s⁻¹z) r

z(z−r)dzE + 1

2π D

u, χ_z(s⁻¹z) r

sz(z−r)idz∧dzE . Proof. Denote byδ_ξ the Dirac mass at ξ∈C. Observe that

i 2π∂ dz

z−ξ =dd^clog|z−ξ|=δ_ξ,

where the equalities are in the sense of currents onC. Hence, for|ξ|< s, i

2π∂hχ(s⁻¹z)dz z−ξ

i

= χ_z(s⁻¹z)idz∧dz

2πs(z−ξ) +χ(s⁻¹z)δ_ξ= χ_z(s⁻¹z)idz∧dz 2πs(z−ξ) +δ_ξ. Applying this identity forξ = 0 and ξ=r, and since u(0)−u(r) =hu, δ₀−δ_ri, we obtain

u(0)−u(r) = 1 2π

D u, i∂

h

χ(s⁻¹z) 1

z − 1 z−r

dz

iE

− 1 2πs

D

u, χz(s⁻¹z) 1

z − 1 z−r

idz∧dz E

= 1

2π D

i∂u, χ(s⁻¹z) r

z(z−r)dzE + 1

2π D

u, χ_z(s⁻¹z) r

sz(z−r)idz∧dzE .

The assertion is proved.

The following is a main result in this section. It will be a crucial technical tool in the construction of the norms with respect to which the transfer operator has a spectral gap.

Proposition 2.11. Let u:B^k₅ →R be continuous and such that k∂uk_L2(B^k₅)<∞. Assume that i∂u∧∂u≤dd^cv where v:B^k₅ →R is continuous, p.s.h., and such that

(2.2)

Z 1 0

m_B^k

4(v, t)(log log^?t)⁴t⁻¹dt <+∞.

Then there is a positive constantc such that, for all 0< r≤1/2, we have m_Bk

1(u, r)≤c

Z r^1/2

0

m_Bk

4(v, t)(log log^?t)²t⁻¹dt 1/2

(2.3)

+cm_Bk

4(v, r)^1/3Ω_Bk

4(v)^1/6(log^?r)^1/2+cΩ_Bk

4(v)^1/2r^1/2(log^?r)^1/2. Proof. Let x, y ∈ B^k₁ be such that dist(x, y) ≤ r ≤ 1/2. We need to bound |u(x)−u(y)| by the RHS of (2.3). We can assume without loss of generality that dist(x, y) =r. By a change of coordinates and restricting to the complex line through x and y we can reduce the problem to the case of dimension 1. More precisely, we can assume that x = 0 and y = r in C and that u, v are defined on D4. By subtracting constants, we can assume that v(0) = 0 and R

D3u(z)idz∧dz = 0. By multiplying u and v by suitable constants γ and γ², we can assume thatm_D3(v,1) = 1/8, which implies that|v| ≤1 on D₃. In order to establish (2.3) it is enough to show that|u(0)−u(r)|is bounded by a constant times

(2.4) Z r^1/2 0

m_D3(v, t)(log log^?t)²t⁻¹dt1/2

+m_D3(v, r)^1/3(log^?r)^1/2+r^1/2(log^?r)^1/2. Sincev is bounded, by Chern-Levine-Nirenberg’s inequality [CLN69] the mass ofdd^cv onD₂ is bounded by a constant. Thus, by the hypotheses on u and v, the L²-norm of ∂u on D2 is

10

bounded by a constant and therefore, by Poincar´e-Wirtinger’s inequality, kuk_W1,2(D2) is also bounded by a constant.

Fix a smooth function 0≤χ(z)≤1 with compact support inD2 and such thatχ= 1 onD1. Define χ_z :=∂χ/∂z and set

s:=rmin

r^−1/2, m_D3(v, r)^−1/3 . We have

(2.5) √

2r≤s≤r^1/2<1

because mD3(v, r) ≤ mD3(v,1) = 1/8 and 0 < r ≤ 1/2. The functions u and χ satisfy the assumptions of Lemma 2.10. Thus, (2.1) holds for the above s and r. The second term in the RHS of (2.1) is an integral over D2s\Ds because χz has support in D2\D1. Moreover, for z ∈ D_2s\D_s, we have _sz(z−r)^r = O(rs⁻³) because of (2.5). Using that kuk_W1,2(D2) . 1 and that Leb(D_2s) . s², Lemma 2.9 implies that the considered term has modulus bounded by a constant times

rs⁻¹(log^?s)^1/2.max

r^1/2, m_D3(v, r)^1/3 (log^?r)^1/2. The last expression is bounded by the sum in (2.4).

In order to conclude, it remains to bound the first term in the RHS of (2.1). Choose a smooth decreasing function h(t) defined for t >0 and such that h(t) := (−logt)(log(−logt))² for t small enough and h(t) = 1 for t large enough. Define η(z) := h(|z|) +h(|z−1|). We will also use the function ˜v(z) :=v(z)−r⁻¹v(r)<(z). This function satisfies dd^cv˜=dd^cv and

˜

v(0) = ˜v(r) = 0. By Cauchy-Schwarz’s inequality we have for the first term in the RHS of (2.1)

D

i∂u, χ(s⁻¹z) r z(z−r)dz

E

2 ≤

i∂u∧∂u, χ(s⁻¹z)η(r⁻¹z)

Z χ(s⁻¹z) η(r⁻¹z)

r²

|z²(z−r)²|idz∧dz.

Using the change of variable z 7→ rz, the fact that 0 ≤χ ≤ 1, and the definition of η we see that the last integral is bounded by

Z

C

idz∧dz h(|z|) +h(|z−1|)

|z²(z−1)²|·

Using polar coordinates forz and for z−1 and the definition of h it is not difficult to see that the last integral is finite. Therefore, sincei∂u∧∂u≤dd^cv=dd^c˜v, we get

D

i∂u, χ(s⁻¹z) r z(z−r)dz

E

2

.

dd^cv, χ(s˜ ⁻¹z)η(r⁻¹z) . Define ˆv(z) := ˜v(sz). The RHS in the last expression is then equal to

dd^cv, χ(z)η(rˆ ⁻¹sz)

=

dd^cv, χ(z)h(rˆ ⁻¹s|z|) +

dd^cv, χ(z)h(|rˆ ⁻¹sz−1|) .

In order to conclude the proof of the proposition, it is enough to show that each term in the last sum is bounded by a constant times

(2.6)

Z s

0

m_D3(v, t)(log^?|logt|)²t⁻¹dt+m_D3(v, r)^2/3log^?r.

We will only consider the first term. The second term can be treated in a similar way using the coordinate z⁰ :=z−rs⁻¹. Since h is decreasing, the first term we consider is bounded by

dd^cv, χ(z)h(|z|)ˆ .

Claim. We have

(2.7)

dd^cˆv, χ(z)h(|z|)

= Z

D2\{0}

ˆ

v(z)dd^c[χ(z)h(|z|)].

11

We assume the claim for now and conclude the proof of the proposition. Notice that the assumption (2.2) will be used in the proof of this claim.

Using the definitions ofh and χ, we can bound the RHS of (2.7) by a constant times Z

D2\{0}

|ˆv(z)z⁻²|(log log^?|z|)²idz∧dz = Z

D2s\{0}

|˜v(z)z⁻²|(log log^?|z/s|)²idz∧dz

. Z

D2s\{0}

|˜v(z)z⁻²|(log log^?|z|)²idz∧dz,

where we used the change of variable z 7→ sz and the fact that log^?|z/s| . log^?|z| for 0 < |z| < 2s < 2. Moreover, by the definition of ˜v and using that v(0) = ˜v(0) = 0, we have for |z|<2s

|˜v(z)| ≤m_D3(v,|z|) +r⁻¹|v(r)<(z)| ≤m_D3(v,|z|) +m_D3(v, r)r⁻¹|z|.

Therefore, using polar coordinates, we see that the last integral is bounded by a constant times Z 2s

0

mD3(v, t)(log log^?t)²t⁻¹dt+mD3(v, r)r⁻¹s(log log^?(2s))².

The first term in this sum is bounded by a constant times the integral in (2.6) because m_D3(v, t⁰) ≤4m_D3(v, t) for s/2 ≤ t ≤s ≤ t⁰ ≤2s. The second one is bounded by a constant times the second term in (2.6) by the definition ofsand (2.5). The proposition follows.

Proof of the claim. Observe thath(|z|) tends to infinity when ztends to 0. Let ϑ:R→R be a smooth increasing concave function such thatϑ(t) =tfort≤0 andϑ(t) = 1 for t≥2. Define ϑn(t) :=ϑ(t−n) +n. This is a sequence of smooth functions increasing to the identity. Define l(z) :=χ(z)h(|z|). Using an integration by parts, we see that the LHS of (2.7) is equal to

n→∞lim

dd^cv, ϑˆ _n(l(z))

= lim

n→∞

Z

D3

ˆ

v(z)dd^cϑ_n(l(z))

= lim

n→∞

Z

D3

ˆ

v(z)ϑ⁰_n(l(z))dd^cl(z) + lim

n→∞

Z

D3

ˆ

v(z)ϑ⁰⁰_n(l(z))dl(z)∧d^cl(z).

The first term in the last sum converges to the RHS of the identity in the claim using Lebesgue’s dominated convergence theorem and (2.2). We need to show that the second term tends to 0.

Since χ(z) = 1 for z near 0, for n large enough, the considered term has an absolute value bounded by a constant times

n→∞lim Z

{h(|z|)>n}

|ˆv(z)|i∂h(|z|)∧∂h(|z|). lim

n→∞

Z

{h(|z|)>n}

|ˆv(z)z⁻²|(log log^?|z|))⁴idz∧dz.

Using the arguments as at the end of the proof of Proposition 2.11 and the assumption (2.2) on v, we see that the last integrand is an integrable function on D1. Since the set{h(|z|)> n}

decreases to{0}whenntends to infinity, the last limit is zero according to Lebesgue’s dominated

convergence theorem. This ends the proof of the claim.

Corollary 2.12. Let S0 be a positive closed (1,1)-current onP^k whose dynamical potentialuS

satisfies ku_Sk_logp ≤ 1 for some p > 1. Let F(S₀) denote the set of all continuous functions g: P^k → R such that i∂g ∧∂g¯ ≤ S0. Then for any positive number q < min ^p−1₂ ,^p₃

we have kgk_logq ≤ c for some positive constant c = c(p, q). In particular, the family F(S₀) is equicontinuous.

Proof. Notice that (2.2) is satisfied for allv such that kvk_logp <∞ for some p > 1. It follows that if u and v are as in Proposition 2.11 and v is log^p-continuous for some p > 1 then u is log^q-continuous on B^k₁ for all q as in the statement, with kuk_Bk

1,log^q ≤ ckvk^1/2

B^k₅,log^p for some positive constant c independent of u, v. The result is thus deduced from Proposition 2.11 by means of a finite cover ofP^k, in the same way as in the proof of Corollary 2.8.

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3. Some semi-norms and equidistribution properties

In this section we consider the action of the operator (fⁿ)∗ on functions and currents. We also introduce the semi-norms which are crucial in our study. Some results and ideas here are of independent interest. Recall that we always assume that f satisfies the Assumption (A)in the Introduction.

3.1. Bounds on the potentials of (fⁿ)∗ωFS. We start by giving estimates on the potentials of the currents (fⁿ)∗ωFS. As explained in the Introduction, these estimates will allow us to globally control the distortion of fⁿ. Define

ω_n:=d^−(k−1)n(fⁿ)∗ωFS.

Recall that f∗ multiplies the mass of a positive closed (1,1)-current by d^k−1. Therefore, all currentsωn have unit mass. We denote byun the dynamical potential of ωn. In particular,u0

is the dynamical potential ofωFS. It is known thatu₀ is H¨older continuous, see [Kos97, DS10a].

Observe thatd⁻¹f^∗ωFS is a smooth positive closed (1,1)-form of mass 1. Therefore, there is a unique smooth function v such that

dd^cv=d⁻¹f^∗ωFS−ωFS and hµ, vi= 0.

Lemma 3.1. We have

un=d^−(k−1)n(fⁿ)∗u0 and u0 =−

∞

X

n=0

d⁻ⁿv◦fⁿ.

Proof. We prove the first identity. Denote byu⁰_n the RHS of this identity, which is a continuous function. By the definition of u_n and the invariance of T, we have

dd^c(u_n−u⁰_n) = (ω_n−T)−d^−(k−1)n(fⁿ)∗(ωFS−T) = (ω_n−T)−(ω_n−T) = 0.

Therefore, u_n−u⁰_n is pluriharmonic and hence constant on P^k. Moreover, the invariance of µ implies that

hµ, u⁰_ni=d^−(k−1)nh(fⁿ)^∗µ, u₀i=dⁿhµ, u₀i= 0.

By the definition of u_n, we also have hµ, u_ni = 0. We deduce that u_n =u⁰_n, which implies the first identity in the lemma.

It is clear that the sum in the RHS of the second identity in the lemma converges uniformly.

Therefore, this RHS is a continuous function that we denote by u⁰₀. The invariance of µ also implies thathµ, u⁰₀i= 0. A direct computation gives

dd^cu⁰₀= lim

N→∞

−

N−1

X

n=0

d⁻ⁿdd^c(v◦fⁿ)

= lim

N→∞ωFS−d^−N(f^N)^∗ωFS =ωFS−T,

where the last identity is a consequence of the definition of T. Since dd^cu0 is also equal to ωFS−T, we obtain that u₀−u⁰₀ is constant onP^k. Finally, using that

hµ, u₀i=hµ, u⁰₀i= 0,

we conclude that u₀ =u⁰₀. This ends the proof of the lemma.

In the sequel, we will need explicit bounds on the oscillation Ω(u_n) ofu_n. These are provided in the next result.

Lemma 3.2. For every constantA >1, there exists a positive constantcindependent of nsuch thatku_nk_∞≤cAⁿ andΩ(un)≤cAⁿ for all n≥0.

13